The alignment conjecture for the padded permanent and determinant

Let $f$ be the form under consideration, let $f_0$ be its leading form, let $L(\lambda)$ be the Levi subgroup associated with the special one-parameter subgroup $\lambda$, and let $H_Y$ be the relevant stabilizer in $GL(Y)$. An alignment between $f_0$ and a conjugate $f^g$ means the alignment relation defined in the paper. Alignment conjecture. If $f_0$ is $L(\lambda)$-stable and is the only form in $\operatorname{Sym}^n(Y)$ with stabilizer $H_Y$, then there is an alignment between $f_0$ and a conjugate $f^g$ of $f$. This is motivated by the proposed connection between stability, uniqueness of the stabilizer form, and alignment in geometric complexity theory; the supplied text gives no resolution status.